Recently, some sufficient conditions for a digraph to have maximum connectivity or high superconnectivity have been given in terms of a new parameter which can be thought of as a generalization of the girth of a graph. In this paper similar results are derived for bipartite digraphs and graphs showing that, in this case, all the known conditions can be improved. As a corollary, it is shown that...

In order to motivate our results we will begin by describing two classical expansions of the chromatic polynomial. Let G be a simple graph with finite vertex set V . Let χG(x) be the number of proper colorings of G with x colors (assignments of colors 1, 2, . . . , x to the vertices of G so that no two adjacent vertices have the same color). If e is an edge of G, let G\e be G with e removed, an...

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

Let G be a simple graph of order n and size m. An edge covering of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Here we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G, x) = ∑m i=1 e(G, i)x , where e(G, i) is the number of edge covering sets of G of size i. Let G and H be two graphs of order n s...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Concl t THEN Lemma `fs unroll`THENM ReduceConcl ;;* THM fs add card>> 8A:D. 8s:FS(A). 8x:|A|. :(x fAg s) => |s +fAg x| = |s|+1 in Int* THM fs del exists>> 8A:D. 8s1:FS(A). 8x:|A|. 9s2:FS(A).(8y:|A|. y fAg s2 <=> y fAg s1 & :(y=x in |A|))& (x fAg s1 => |s2| = |s1|-1)* THM fs del>> A:D -> FS(A) -> |A| -> FS(A)Extraction:A s x. fs del exists(A,s,x).1...

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